Question

Xn bounded above. Show that sup(x_k, so k ≥ n) is bounded

a) The sequence is unbounded
b) The supremum is always zero
c) The sequence is bounded
d) The supremum is undefined

Answer 1

To show that sup(x_k) is bounded, we need to prove that the sequence (x_k) is bounded. the correct option is c) The sequence is bounded.

Step-by-step explanation:

To show that sup(x_k) is bounded, we need to prove that the sequence (x_k) is bounded. A sequence (x_k) is bounded if there exists a real number M such that |x_k| < M for all k. Given that Xn is bounded above, this means that there exists a real number N such that Xn < N for all n.

Now, consider the sequence sup(x_k, for k ≥ n). Since Xn is bounded above, we know that Xn < N for all n. Therefore, the elements of the sequence sup(x_k, for k ≥ n) will be less than or equal to N for all n.

This implies that the sequence sup(x_k, for k ≥ n) is bounded above by N, which in turn means that sup(x_k) is bounded. Hence, the correct option is c) The sequence is bounded.